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BinomialSum

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Description

Summary

A summation over k from 0 to N of the binomial coefficient
N! / (k! (N - k)!) times a power or a trigonometric weight, in closed form:
sum(N! / (k! (N - k)!) x^k, k, 0, N) is (1 + x)^N, and
sum(N! / (k! (N - k)!) cos(k t), k, 0, N) is (2 cos(t/2))^N cos(N t / 2).

Remarks

The binomial theorem read backwards. With weights x^k y^(N-k) the sum is
(x + y)^N, either power optional. With cos(k t) it is the real part of
(1 + e^(i t))^N, and 1 + e^(i t) = 2 cos(t/2) e^(i t/2) exactly, so the sum is
(2 cos(t/2))^N cos(N t/2); with sin(k t) the same with the sine. Both hold for
every complex t, since cos z = (e^(iz) + e^(-iz)) / 2 is the definition
and the two conjugate sums add: no assumption on t is owed.
 The one condition is the range: for N below zero the summation is empty and this
library answers an empty range with 0, while the formula does not, so a symbolic
N gets the same piecewise PolynomialSummation attaches. Recognised
structurally, on the factors of the simplified summand: the three factorials with
N the upper bound as written, at most one power in k, at most one in
N - k, at most one cosine or sine of k times something free of it, and
nothing else that mentions k. A trigonometric weight beside a power is declined:
the closed form then needs the modulus and argument of y + x e^(i t), which is
not a simplification. Question II.3 of
#1212.

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